The above definition is very general. Many theorems, and in particular regularity theorems, require extra assumptions in the kernels $K$. These assumptions restrict the study to certain sub-classes of linear operators. The simplest of all is the fractional Laplacian. We list below several extra assumptions that are usually made.

The fractional Laplacian

The fractional Laplacian is the simplest and most common purely integro-differential operator. It corresponds to a translation invariant operator for which $K(y)$ is radially symmetric and homogeneous.

Stable operators

These are the operators whose kernel is homogeneous in $y$
\[ K(y)=\frac{a(y/|y|)}{|y|^{n+s}}\qquad\textrm{or}\qquad K(x,y)=\frac{a(x,y/|y|)}{|y|^{n+s}}.\]
They are the generators of stable Lévy processes. The function $a$ cound be any $L^1$ function on $S^{n-1}$, or even any measure.

Uniformly elliptic of order $s$

This corresponds to the assumption that the kernel is comparable to the one of the fractional Laplacian of the same order.
\[ \frac {(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac {(2-s)\Lambda}{|y|^{n+s}}. \]

The normalizing factor $(2-s)$ is a normalizing factor which is only important when $s$ approaches two.

An operator of variable order can be either one for which $s$ depends on $x$, or one for which there are two values $s_1<s_2$, one for the left hand side and another for the right hand side.

Order strictly below one

If a non symmetric kernel $K$ satisfies the extra local integrability assumption
\[ \int_{\R^n} \min(|y|,1) K(x,y) \mathrm d y < +\infty, \]
then the extra gradient term is not necessary in order to define the operator.

\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x)) \, K(x,y) \mathrm d y. \]

The modification in the integro-differential part of the operator becomes an extra drift term.

A uniformly elliptic operator of order $s<1$ satisfies this condition.

Order strictly above one

If a non symmetric kernel $K$ satisfies the extra integrability assumption on its tail.
\[ \int_{\R^n} \min(|y|^2,|y|) K(x,y) \mathrm d y < +\infty, \]
then the gradient term in the integral can be taken global instead of being cut off in the unit ball.

This last assumption is a technical restriction which measures the contribution of the non-symmetric part of $K$ and is only relevant for the limit $\alpha \to 1$.

Indexed by a matrix

In some cases, it is interesting to study a family of kernels $K$ that are indexed by a matrix. For example, given the matrix $A$, one can consider the kernel of order $s$:
\[ K_A(y) = \frac{(2-s) \langle y , Ay \rangle}{|y|^{n+2+s}}. \]
This family of kernels has the outstanding property that the corresponding linear operator $L$ coincides with $Lu(x) = a_{ij} \partial_{ij}\left[(-\Delta)^{(s-2)/2} u \right] (x)$ for some coefficients $a_{ij}$. This property has been exploited in order to obtain an ABP estimate in integral form. [2]

Second order elliptic operators as limits of purely integro-differential ones

A class of kernels that is big enough to recover all translation invariant elliptic operators of the form $Lu(x) = Tr ( A \cdot D^2u(x) )$ is given by the kernels
\[ K_A(y) = (2-\sigma) \frac{1}{|Ay|^{n+\sigma}},\]
where $A$ is an invertible symmetric matrix.

Conversely, the condition
\[ \lambda |e|^2 \leq \int_{\partial B_1} (x \cdot e)^2 a(x) \, \mathrm d S(x) \leq \Lambda |e|^2, \qquad \text{for all vectors } e \in \R^d,\]
defines the largest class of stable operators that may be considered uniformly elliptic. Indeed, this is the condition that ensures regularity of solutions to translation invariant integro-differential equations.[3] If we let the operator depend on $x$, it is an outstanding open problem whether this condition alone (for every value of $x$) ensures Holder estimates for the integro-differential equation.

Characterization via global maximum principle

A bounded linear operator

\[ L: C^2_0(\mathbb{R}^n) \to C(\mathbb{R}^n) \]

is said to satisfy the global maximum principle if given any $u \in C^2_0(\mathbb{R}^n)$ with a global maximum at some point $x_0$ we have

\[ (Lu)(x_0) \leq 0 \]

It turns out this property imposes strong restrictions on the operator $L$, and we have the following theorem due to Courrège [4][5]: if $L$ satisfies the global maximum principle then it has the form